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It is not possible to bound the H s norm at a given time t0 > 0 in terms of Sobolev norms of the initial data.

Applications of the theory of evolution equations to general relativity

The group norms deﬁned in this section are different from the group algebra norms deﬁned in Section 5.

Quantum Hidden Subgroup Problems: A Mathematical Perspective

All through the paper, | · | denotes the euclidean norm on Rd, and |ℓ|1 = Pi |ℓi | the ℓ1 -norm.) Note that for ζ = 0, this is just a usual half-space.

A law of large numbers for random walks in random mixing environments

Let k · kE be a 1-unconditional norm on RN, by which we mean that the standard basis of Rn is a 1-unconditional basis for k · kE (see ; such a norm is also sometimes called absolute).

Concentration of norms and eigenvalues of random matrices

In particular, as long as one has the appropriate Lipschitz estimates, the underlying normed spaces need not be unconditional, nor must the norm on matrices even be an operator norm.

Concentration of norms and eigenvalues of random matrices

We remark that when p = q ′, the l.h.s. of (11) was shown by Persson to coincide with both the q -summing norm πq (T ) and the q -nuclear norm νq (T ) of the kernel operator T : ℓq ′ → ℓq given by T (ej ) = P∞k=1 cj k ek .

Concentration of norms and eigenvalues of random matrices

Note that under this identiﬁcation, the norm k · ke on Er,n corresponds to the usual Euclidean norm on Rrn2 .

A new application of Random Matrices: Ext(C*_{red}(F_2)) is not a group

The following theorem states that the imbedding of L1 (D) ∩ SA,C (D) (with the L1 norm) into LA ∩ SA,C (with the LA norm) is continuous.

Random Surfaces

The vacuum modes A1 and A2 are normalized according to Eq. (2.7) with the scalar product (7.20), taking into account that both AR and AL are normalized and orthogonal, and that the norm of A∗ is the negative norm of A.

Quantum Physics of Simple Optical Instruments

The semi-norms in the basic energy estimate (13) now depend on ε ∈ (0, 1], and, upon applying Gronwall’s inequality, will appear as exponents in the L2 -norm estimates of a prospective generalized solution; this suggests to assume logarithmic bounds on the symbols.